A counterexample to the monotone increasing behavior of an Alt-Caffarelli-Friedman formula in the Heisenberg group

TL;DR

This paper presents a counterexample demonstrating that the monotone increasing behavior of a specific Alt-Caffarelli-Friedman related function does not hold in the Heisenberg group, challenging previous assumptions.

Contribution

It provides the first known counterexample to the monotonicity of the Alt-Caffarelli-Friedman formula in the noncommutative Heisenberg group setting.

Findings

Counterexample disproves monotonicity in Heisenberg group

Challenges previous assumptions about formula behavior

Highlights differences between Euclidean and noncommutative frameworks

Abstract

In this paper we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced in \cite{FeFo}, companion of the celebrated Alt-Caffarelli-Friedman monotonicity formula, in the noncommutative framework.